It is well known that in the quantization of certain relativistic theories such electromagnetism or relativistic string negative norm states could arise when quantizing covariantly. Acting with creation and annihilation operators over the ground state one can construct the state space. Some of these have negative norm but when we impose the constraint ghosts disappear.

For the relativistic particle I am lookng for a similar idea to construct states with negative norm, then the constraint $p^2+m^2=0$ should eliminate them. Certainly this case is different because you dont have oscilator operators but I guess that ghost should appear too. I suspect that the eigenvectors of $P_0$ could have negative norm but I haven't been able to prove it.

I was looking for reference or books treating this but nothing was found, the only book mentioning a little bit is Introduction to Superstrings and M-theory by Michio Kaku in page 29.


I would like to underline the fact that my question is not about brst quantization. Do not talk me about the ghost and antighost.

I just want to know :

(1)if eigenvectors of $P_0$ have negative norm.

(2)what states kill the constraint $p^2+m^2$. In a similar way that in string theory the virasoro constraint kills ghosts and the gauss law kills ghost photon states in electromagnetism. Note that Kaku in page 29 says that this is a "ghost killing constraint" in what he calls Gupta-Bleuler quantization.

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    $\begingroup$ Take a look at this article: BRST quantization of point particle $\endgroup$ – Prof. Legolasov Jun 2 '15 at 14:35
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    $\begingroup$ I'm not quite sure what you want - do you want to see how one explicitly performs the (BRST) quantization of the point particle? Do you want to know whether there is an "ad hoc" method similar to Gupta-Bleuler quantization which does not explicitly use the full BRST formalism for constrained systems? I think the coordinates $X^\mu$ of the point particle are essentially just a massive vector field from an abstract point of view, and the constraint $p^2 + m^2=0$ leads to the spurious zero norm states that decouple, but no negative norm states. $\endgroup$ – ACuriousMind Jun 2 '15 at 15:32
  • $\begingroup$ nothing to do with brst quantization. I am talking about the examples of electromagnetism and simple bosonic string. you can see in any textbook that ghosts arise when one wants to impose a covariant communtation rule $[x,p]=i\hbar$ and they are not using brst quatization. So your response would be that there are no negative norm states? $\endgroup$ – Anthonny Jun 2 '15 at 18:37
  • $\begingroup$ Kaku called it Gupta-Bleuler quantization, and in other books it is called Old covariant quantization. $\endgroup$ – Anthonny Jul 25 '15 at 19:04

If $|p\rangle$ is an eigenstate of $p^\mu$,there is no reason to make this state a ghost. Ghost would appear if you have polarization in this states. Is better to see this in a second quantization. Now, you have an operator $a_p^{\dagger}$ creating momentum and an operator $a_x$ for the position. Then $$ [x^{\mu},p^\nu]=i g^{\mu\nu} \rightarrow x^\mu p^\nu[a^\dagger_x a_x,a^\dagger_p a_p]=ig^{\mu\nu} $$ and the usual commutation relation for the creation and annihilation operators. Note that $a_p$ and $a_x$ are scalars and the usual commutation apply. There is no ghost.

The ghost arises when you have Lorentz indices in the creation and annihilation operators. In others words, ghost arises from the incompatibility between the finite representations indicies and the litlle group indices.

In the case of the scalar particle, the coinstraint only work to fix the mass of the particle. For an arbitrary state, if you apply a lorentz transformation you would see a superposition of masses and signal of $p^0$. Is like a superposition between species of particles. You would see tachyons too.

The mass of the particle is an invariant under lorentz transformations. So is better to separate particles with different masse if you want to create an interacting theory that preserves this symmetry.

In the case of particles that carries polarization, ghost is expected because of the time-like polarizations. And you need extra constraint to eliminate them.


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